Real numbers

All four operations for the integers (+,-,*,/)

Pencil and paper algorithms for all four operations

Rational numbers

The real numbers are defined as the unique complete ordered field. A field is defined as a set of numbers such that:

Given any three numbers, the order of addition doesn't matter (e.g., 2+(3+5)=(2+3)+5)

Given any three numbers, the order of multiplication doesn't matter (e.g. 2(3(5))=(2(3))5)

Given any two numbers, addition doesn't care about order (e.g., 2+3=3+2)

Given any two numbers, multiplication doesn't care about order (e.g., 2(3)=3(2))

Adding zero and multiplying by one preserve the number, and 0 isn't equal to 1 (e.g., 2+0=2(1)=2)

Every number can be subtracted from 0 (e.g. 0-2=-2 and 2+(-2)=0)

1 can be divided by any nonzero number (e.g., 1/2=1/2 and 2(1/2)=1)

The product of a sum is the sum of the products (e.g., 2(3+5)=2(3)+2(5))

An ordered field is a field where the numbers can be ordered such that:

Addition preserves the ordering (e.g. since 2<3, 2+5<3+5)

Multiplying two numbers greater than zero (i.e., two positive numbers) gives a positive number (e.g., since 2>0 and 3>0, 2(3)>0)

A complete ordered field is an ordered field such that every set of numbers bounded from above has a least upper bound.

Look at the place-value chart here:

Notice how it stopped in the ones place? What if it didn't stop? What if we have something like this instead:

You would get all nonnegative real numbers.

There is a caveat, however; every nonzero terminating real number has an alternate expansion with infinitely many nines. For example, since 1=9/10+9/100+9/1000..., 1=0.9

Addition, subtraction, multiplication and division all use the same method as integers. However, there are two things that should be noted.

Addition, subtraction and multiplication work by starting at the last digit. But in the real numbers, there sometimes is no last digit. So, to multiply 0.583436946468... by 3.525534690266..., you have to do something else. You have to do 0.5*3.5, 0.58*3.52, 0.583*3.525 etc. to see what happens at infinity.

In division, you don't stop at the ones place and take the remainder. Instead, you keep going forever.

All rational numbers are real numbers. This can be shown by performing the division described by the rational number. For instance, 3/7=0.428571. Are all real numbers rational? No. For we can do this:

Since the list contains all rational numbers from 0 to 1 (exclusive), the new number is between 0 and 1 (exclusive), and the new number isn't on the list (as it differs from each listed number in at least one position - the shaded digits), the new number is irrational. This also shows that there are more real numbers than natural numbers (if there are as many of the latter as the former, a list could be made containing all real numbers, then the same trick can be applied to arrive at a contradiction)

• Wikipedia: Real number